(1+i)^12 Polar Form

Jasonbradley

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Jun 16, 2024

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Finding the Polar Form of (1 + i)^12

This article explores the process of finding the polar form of the complex number (1 + i)^12. We'll delve into the steps involved, utilizing concepts like magnitude, angle, and De Moivre's Theorem.

Understanding Polar Form

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ) relative to the positive real axis on the complex plane. The general form is:

z = r(cos θ + i sin θ)

Steps to Find the Polar Form

1. Finding the Magnitude (r):

   • The magnitude of a complex number z = a + bi is calculated as: r = √(a² + b²)

   • For (1 + i), r = √(1² + 1²) = √2

2. Finding the Angle (θ):

   • The angle θ is found using the arctangent function: θ = arctan (b/a)

   • For (1 + i), θ = arctan (1/1) = 45° (or π/4 radians). Since (1 + i) lies in the first quadrant, this angle is correct.

3. Applying De Moivre's Theorem:

   • De Moivre's Theorem states: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

   • Applying this to (1 + i)^12:

     • (√2 * (cos 45° + i sin 45°))^12
     • = (√2)^12 * (cos (12 * 45°) + i sin (12 * 45°))
     • = 2^6 * (cos 540° + i sin 540°)

4. Simplifying the Angle:

   • Since the cosine and sine functions have a periodicity of 360°, we can simplify 540°:
   • cos 540° = cos (540° - 360°) = cos 180° = -1
   • sin 540° = sin (540° - 360°) = sin 180° = 0

5. The Final Polar Form:

   • Therefore, the polar form of (1 + i)^12 is:
     • 2^6 (-1 + 0i) = -64

Conclusion

The polar form of (1 + i)^12 is -64, which represents a complex number with a magnitude of 64 and an angle of 180° (or π radians) on the complex plane. This process demonstrates the power of De Moivre's Theorem in simplifying complex number operations.